A quantitative analysis of Galilei’s observations of Jupiter satellites from the Sidereus Nuncius (2025)

3.1 Conversion of the observation times

Galilei expressed the observation times in terms of hours since sunsetfollowing italic hours that was used at those times. The first hour, for example, refers to one hour after sunset. To convert these data into modern hours (CET) we considered the time of sunset in Padua inthe beginning of the year (see Fig.86, Appendix) andsum the hour reported by Galilei to this value assuming that each hourhas the same length. The time of sunset was approximate linearly as:$h_{sunset}=16.58+0.02353~{}d$ where hours are expressed in decimalfractions and $d$ is the ordinal day since the beginning of the year. Theresult of the conversion is shown in Tab.3 and 4 where all the observations are associated to a time expressed in CET. It should be noted that 11 times Galilei does not mention the time of the observations (marked as “ïn the column 4 Tab.3 and 4). In this case for the subsequent analysis we assumed that the time was the first hour, a frequent choice. This introduces for these observations an error up to potentially 5-6 hours⁵⁵5The latest observations are before 1 AM. Note that the setting time of Jupiter was about 5.30 and 2.00 at the beginning and end of the considered period. which can be relevant for satellites passing close to Jupiter.

3.2 Satellite-untagged analysis

Before trying to associate the observations to individual satellites,we perform a global analysis putting all the measurementstogether. The disentangling of the contributions of single satellites was not obvious for Galilei as the data were taken sparsely and often satellites were too close and unresolved. The oscillatory pattern of Callisto is nevertheless quite easily separable (see Fig. 3).

We first check if the distribution is consistent with the sum of four harmonic oscillators corresponding to four distinct circular orbits and then consider how the four frequencies might emerge using the Lomb-Scargle algorithm ([4, 5]).

The distribution of the horizontal displacements ($x_{i}$) is shown in Fig.4 for the data (bullets) and the expectation in the hypothesis that the motion is harmonic. As expected the probability of observing the satellite when it is at the maximal elongation is higher as its projected velocity is smaller. Galileo did not arrive to notice that the satellites were moving faster when they were closer to Jupiter which is something that might have been at hand with his data.

The expectation in the solid magenta line assumes that satellites areonly invisible if they are behind or in front of the disk ofJupiter. The expectation is solid green line shows instead thedistribution assuming that the inefficient angular region close toJupiter corresponds to 2.5 times the real dimension of the disk. Itis visible how this second hypothesis is quite consistent with the data.

The superposition of the distribution (that are each normalized to eachother) was done by eye by scaling the $x$-axis to match the mostextreme observations of Callisto. The dimensions of the orbits arebased on modern data. It should be noted that this is very approximateas it assumes that data were taken approximately uniformly in timewhile, as we saw, Galilei was often taking several, near in time,measurements per night. The large statistical error allows to neglectthis and we can say that the observation are semi-qualitativelyconsistent with what we expect. We will perform fully quantitative considerations anywayin the following.

To isolate the outermost satellite, Callisto, one could only select data being more than four diameters away from Jupiter (see Fig.4) where only Callisto can reach. After fitting these data with a sinusoid, the points compatible with the fitted Callisto trajectory could be then removed in the innermost part. There one wouldhave been left with three contributions and iterate. Yet looking at the data it is clear that this was not at all straightforward.

3.3 Lomb-Scargle periodograms

We now try to use the modern technique proposed by Lomb-Scargle[4, 5] to single out the emergence of the expected four distinct mono-chromatic frequencies in the data.

We show in Fig.5 the LS periodogram of the full dataset.

The top plot shows thetime-series ($x_{i}$ vs $t_{i}$, with $t_{i}$ in fraction of years since 1/1/1610) while at the bottom show the corresponding LS periodogram. The python scipy.signal package was used (lombscargle). Vertical coloured lines indicate the value of the expectedfrequencies based on modern data. From left to right, blue isCallisto, green Ganymede, light green Europa and red, Io. It can beseen that peaking structures in the periodogram actually emerge at the correct locations. There is also a spurious one at about 1000 rad/year with a significance similar to the one of the satellites. The peak corresponding toCallisto, is, as expected, the most prominent and with the bestsignal-to-noise ratio while the other three have a weaker evidence. We will later re-apply the algorithm to the satellite-tagged four datasets that will be described in the next section.

3.4 Association of the satellites using a sky simulator

This association allows inspecting the expected pattern of satellitesusing the Stellarium sky simulator.

An example of such an exercise is shown in Fig.6 for the observation number 7, performed on Jan. 15 1610 at the third hour (Galilei’s time) or 19.55 CET. The correspondence is remarkable and this happens for themajority of the cases as it will be shown later (see Appendix A). It is worth noticingalso the level of attention of the recordings where the slightdisplacement of Callisto from the ecliptic plane is reported.

It is interesting to notice that, according to the simulator, the satellites’ magnitudes were, in decreasing brightness: Ganymede (4.75), Io (5.11), Europa (5.43) and Callisto (5.69). Galilei reports instead in decreasing brightness Callisto, Ganymede, Europa, Io. It is possible that the closeness to Jupiter might have played a role in biasing the perception of the brightness⁶⁶6It should also be noted that the accuracy of the predictions of Stellarium have not been cross-checked with other simulators..

Another evident feature is the fact that the size of Jupiter’s disk is generally overestimated by Galilei. This is probably due to the limited resolution of its telescope. This interpretation is corroborated by the fact the often satellites were missed when too close to the planetary disk, as it will be detailed later.

The typographic symbol employed to denote Jupiter is circular but the inner white part seems to hint at the fact the indeed the disk has some equatorial bulging. On the other hand it should be noted that in those days Jupiter was far from opposition so the left limb was in shadow (see for example Fig.6) making it more circular-shaped. Furthermore it should be noticed that the equatorial bulging, is a 7% effect, so it needs an appropriate resolution to be discernible.

Another feature that is apparent already from this single image is that the positions of the outer satellites Ganymede and Callisto seems to be underestimated by Galilei.

Each sketch was put side by side with the output of the Stellariumweb-based sky simulator after taking care to see the values to thosethat were translated in modern time. All the images from Stellariumare shown together with the sketches of Galileo in Appendix A.

The output of the associations is given in Tabs.5 and6. In several cases two satellites were too close to beseparated by Galileo. When this happens we decided to perform theassociation with one of the two. In general we find an almost perfectcorrespondence between the notes of Galileo and the simulator.

In total the satellites that were potentially visible but are notrecorded are 33 (Io 17 times, Europa 8 times, Ganymede 4 times andCallisto 4 times). We have excluded the cases in which the satelliteswere transiting the disk of Jupiter or were behind it. The reason forthe inefficiency of the recordings is almost always due to theproximity of the satellite to the disk of the planet that must havebeen blurry. We can estimate that Galileo could see the satellites isthey were away from the limb of the planet by at least approximately2.5 diameters or about 1’40”.It should be noted that on January 7 1610 Jupiter was 4.28 AU away corresponding to about 45” for the disk angular diameter while on March 1 it was 5.02 AU away corresponding to 38”.This change in angular diameter of the system by 18% is not taken into account in the following analysis as we always report the elongations of satellites normalised to the angular diameter of the disk. Indeed the result suppor the validity of this assumption.

Observation 46 is probably one of the only that is difficult to reconcile with the prediction of the simulator. Galilei sees two satellites on the left and two on the right. It seems that either Io or Europa are on opposite sides of the disk while the simulator predicts both being on the right. We have checked if this could be related to a mismatch in the recorded hour but in the night of Feb 11 the pattern was the same as observation 45 (both Io and Europa left). The transit of both happens almost parallelly during the daytime of Feb 12 and during the early night of 12 they are both on the right side.

For observation 49 it is interesting to note that Callisto is notrecorded even if it is less close than Europa that was seen instead.

3.5 Periodograms of the satellite-tagged datasets

The periodograms of the time series separated by satellite using thesky simulator are shown in Fig.7.

In this case the peaks of each satellite emerge without ambiguity at the expected position. Only for Io there is a less-significant, but still important, peak at about 1000 rad/y that had already been observed in the inclusive analysis of Fig.5. In general the smaller peaks do not occur in correspondence of the expectations for other satellites supporting the fact that the rate of wrong-association is likely very low.

3.6 Sinusoidal fits and orbital parameters extraction

The elongation datasets, ($x_{i}$, $t_{i}$), separated by satellite have been fitted with a function of the form $x(t)=A\sin(\omega(t-t_{0})+\phi_{0})$ using the ROOT [6] fitter libraries. We have set the uncertainty on the point to match the r.m.s. of the residuals in practice setting the to the errors determined by the fit itself.The data with the superimposed best fita are shown in Fig.8.

The same results are shown all together in Fig.9.

A video showing the result of the sinusoidal fits and the satellites’ associations is available in [7]. There the filled points show the prediction of the fit. At the time of observation the animation is briefly pause and the measurements fromGalilei are superimposed together with an image of the original sketch.

The residuals of the fit ($x_{i}-x(t_{i})$) are shown for each satellite in Fig.10. The top row shows them as a function of the point time while the bottom rows shows their distribution fitted with a Gaussian model.

The model is reproducing the data very well without particular trends in the residuals or anomalies. In terms of the Jupiter disk in Galilei’s sketches (GJD) The r.m.s. of the residuals is 0.5 GJD for Io, 0.4 GJD for Europa, 0.56 GJD for Ganymede and 0.63 GJD for Callisto. Considering an average angular disk size of 42” during the measurements, and a factor 2.5 between the real and reported angular size, the previous r.m.s. correspond to values ranging from 40” to about 70”. This can be considered as an estimate of the pointing accuracy of the observations.

In the following we try to imagine how these data can be used to test the validity of Kepler third law for the Jupiter system and the 1:2:4 Laplace resonancebetween the periods of Io, Europa and Ganymede.

To do this test we have converted the radiuses of satellites’ orbits with a scaling factor chosen such that the scaled value for Ganymede corresponds to the modern measured value of 1.0704million of km⁷⁷7The scaling factor is 1.0704Gm/$(4.482\pm 0.084)$GJD = 2.39$\times$10⁸m/GJD..

Figure 11 and Tab.7 show a comparison between the values of the satellites’ orbits radiuses $A$ and their periods $T=2\pi/\omega$ as found by Galilei (bullets) and by modern determinations (histogram).

On the $x$-axis we show the satellite identifier (from left to right Io, Europa, Ganymede, Callisto). Periods are in seconds, are shown in the top left plot while radiuses (in meters) in the top right plot. The bottom plots show the ratio between the measured value by Galilei and the modern value. The periods are compatible with modern values within their errors that range from 0.1% to 0.3%. As far as the radiuses are concerned the ratios are compatible with unity with 1-2 $\sigma$ for Io and Europa. The one of Ganymede is one by construction while for Callisto $A$ is underestimated by about 12%. The statistical errors on these parameters range from 1.9% for the outer satellites to 4.3% for Io. There is hence a trend for a significant underestimation of the major semi-axes for Callisto (See Fig.12, bottom right plot).

Figure 12 (left) shows a scatter plot of$A$ vs $T$. Superimposed a curve of the type $T^{\frac{2}{3}}$. The curve is the function: $A(T)=k_{J}T^{\frac{2}{3}}=\left(\frac{GM_{J}}{4\pi^{2}}\right)^{\frac{1}{3}}T^$ with$G=6.67430\times 10^{-11}$ Nm²kg^-2 and $M_{J}=1.898\times 10^{27}$kg hence $A(T)=5.02\times 10^{5}\times T^{\frac{2}{3}}$.

In determining the law in physical units we have used our knowledge on the orbit of Ganymede to pass from angles to distances. It is interesting to notice that, if you use the knowledge on the distance of the Earth and Jupiter from the Sun, this measurement allows determining the ratio between the mass of Jupiter and that of the Sun just by comparing $k_{J}$ and $k_{Sun}$. Still you have to know that the Kepler constants depend on the mass to one third which is something that will not be clear until Newton’s law.

In Fig.13 we compute the ratios $T_{Eu}/T_{Io}-2$ and $T_{Eu}/T_{Io}-4$ to verify the level of accuracy in determining the orbital resonance effects. Black filled bullets indicate the measurements of Galilei and the blue hollow ones the modern values. Good agreement is found. Deviations from an exact 1:2:4 ratios, also present in modern data, are due to the slow precession of Io’s orbit’s closest point to Jupiter (perijove).

3.7 The performances of the Sidereus Nuncius “cannocchiale”

Unfortunately the instrument used in early 1610 has been lost but we have other instruments that were used by Galilei. Some optical tests performed on two of these instruments is available at [8] and considerations on its aberrations in [9].In this section we want to imagine what could have beenthe performances of the telescope based on the accuracy obtained on the Jupiter system. In the following we use sharp images and apply a very simple Gaussian smearing plus a deformation that brings objects close to the edge of the field of view to appear closer to the center than they really are. This is inspired by the observation that orbits of Ganymede and Callisto appear to be underestimated. This should be verified with an optical simulation or with a system that tries to reproduce the original devices. Furthermore the effect could be also simply related to the increased difficulty of correctly estimating the angular separation when the two points have a larger separation separated. Also it is likely that the distances were estimated in steps, and that Jupiter might have not always been in the center of the field. Despite of this we have nevertheless included this effect in the simulation. The idea is to try and get an impression of what would have been the image available to Galilei.

In Fig.14 we show how the two effects act on the Jupiter system.

Applying the same smearing and aberration effects to terrestrial targets allows to get an insightof the impression that the observation with the “cannocchiale” might have given to people of those times. We know that in August 1609, i.e. about five months before the observations of Jupiter, Galilei had demonstrated the potentials of the cannocchiale to the Doge of Venice, Leonardo Donà (Fig.15, left), probably with a telescope that was still not as good as the later one.

Here we assume that the observations were done from the Campanile of San Marco and, for simplicity, we use the performances of the telescope of January 1610. As a target we have chosen a quite peculiar on: the Scala Contarini del Bovolo that is a very distinctive building located at 360m East of the observation point and with a diameter of about 6m (Fig.15, center and right). It would have surely attracted the attention of the viewers from the Campanile.

The close up on the right of Fig.15 is given to show the, very narrow, field of view of the Galileian telescope (circle) that we estimate in about 20” or a third of degree.

The view of the scala is shown in a modern detailed picture in Fig.16 (left) while the other pictures show the effect of resolution (center) and resolution and aberration (right).

Despite the distortions and the narrow field of view it is clear how amazing this could have been and indeed we know how enthusiastically this technological advance was received, granting Galilei an immediate increase in salary.